There is a ferry service between the northern and southern shore of Balaton (between Tihany and Szántód) which leaves every half an hour during summer and every hour in the rest of the year (except December through February).
It costs 400 HUF (1.50 EUR) per person and 1,100 HUF (4.20 EUR) per car.
The more comfortable is of course your own car. But very interesting may be getting there by ferries, which go there from many resorts on Balaton shores (f.e. from Siofok and Balatonfured). There is no railway on the Tihany, but many buses go there (I know that from Balatonfured there are two buses per hour). The another possibility is to rent a bike (again for example from Balatonfured) - comfortable bike paths leads to Tihony.
While in Hungary be careful to say "Autobus" as bus is the Hungarian word for f**k.
Tihany is a good hike, and it is uphill, so I recommend motorized transportation.